20922195133

Committed 12 Jan 2026 02:04PM UTC coverage: 92.391% (-0.05%) from 92.445%

Build # 20922195133

Build Type

Pull #108

github

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web-flow

Commit Message

Merge 3f49c6580 into cc74f1371

Pull Request Pull Request #108: Support for arbitrary element types and sparse matrices

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/src/functionals.jl

export norm_trace, trace_distance, norm_hs, hs_distance,
    purity, fidelity_sqrt, fidelity, gate_fidelity, shannon_entropy,
    vonneumann_entropy, renyi_entropy, relative_entropy, kl_divergence,
    js_divergence, bures_distance, bures_angle, superfidelity, negativity,
    log_negativity, ppt, concurrence

"""

- `A`: matrix.

Return [trace norm](https://www.quantiki.org/wiki/trace-norm) of matrix `A`.
"""
norm_trace(A::AbstractMatrix{<:Number}) = sum(svdvals(A))

"""

- `A`: matrix.
- `B`: matrix.

Return [trace distance](https://www.quantiki.org/wiki/trace-distance) between matrices `A` and `B`.
"""
function trace_distance(A::AbstractMatrix{T1}, B::AbstractMatrix{T2}) where {T1<:Number, T2<:Number}
    norm_trace(A - B) / 2
end

"""

- `A`: matrix.

Return [Hilbert–Schmidt norm](https://en.wikipedia.org/wiki/Hilbert%E2%80%93Schmidt_operator) of matrix `A`.
"""
norm_hs(A::AbstractMatrix{<:Number}) = sqrt(sum(abs2.(A)))

"""

- `A`: matrix.
- `B`: matrix.

Return [Hilbert–Schmidt distance](https://en.wikipedia.org/wiki/Hilbert%E2%80%93Schmidt_operator) between matrices `A` and `B`.
"""
function hs_distance(A::AbstractMatrix{<:Number}, B::AbstractMatrix{<:Number})
    norm_hs(A - B)
end

"""

- `ρ`: matrix.

Return the purity of `ρ` ∈ [1/d, 1]
"""
purity(ρ::AbstractMatrix{<:Number}) = tr(ρ^2)

"""

- `ρ`: matrix.
- `σ`: matrix.

Return square root of [fidelity](https://www.quantiki.org/wiki/fidelity) between matrices `ρ` and `σ`.
"""
function fidelity_sqrt(ρ::AbstractMatrix{<:Number}, σ::AbstractMatrix{<:Number})
  if size(ρ, 1) != size(ρ, 2) || size(σ, 1) != size(σ, 2)
    throw(ArgumentError("Non square matrix"))
  end
  λ = real(eigvals(ρ * σ))
  r = real(sum(sqrt.(λ[λ.>0])))
end

"""

- `ρ`: matrix.
- `σ`: matrix.

Return [fidelity](https://www.quantiki.org/wiki/fidelity) between matrices `ρ` and `σ`.
"""
function fidelity(ρ::AbstractMatrix{<:Number}, σ::AbstractMatrix{<:Number})
  fidelity_sqrt(ρ, σ)^2
end

fidelity(ϕ::AbstractVector{<:Number}, ψ::AbstractVector{<:Number}) = abs2(dot(ϕ, ψ))
fidelity(ϕ::AbstractVector{<:Number}, ρ::AbstractMatrix{<:Number}) = real(ϕ' * ρ * ϕ)
fidelity(ρ::AbstractMatrix{<:Number}, ϕ::AbstractVector{<:Number}) = fidelity(ϕ, ρ)

"""

- `U`: quantum gate.
- `V`: quantum gate.

Return [fidelity](https://www.quantiki.org/wiki/fidelity) between gates `U` and `V`.
"""
function gate_fidelity(U::AbstractMatrix{<:Number}, V::AbstractMatrix{<:Number})
    abs(1.0 / size(U,1) * tr(U'*V))
end

"""

- `p`: vector.

Return [Shannon entorpy](https://en.wikipedia.org/wiki/Entropy_(information_theory)) of vector `p`.
"""
shannon_entropy(p::AbstractVector{<:Real}) = -sum(p .* log.(p))

"""

- `x`: real number.

Return binary [Shannon entorpy](https://en.wikipedia.org/wiki/Entropy_(information_theory)) given by \$-x  \\log(x) - (1 - x)  \\log(1 - x)\$.
"""
shannon_entropy(x::Real) = x > 0 ? -x * log(x) - (1 - x) * log(1 - x) : error("Negative number passed to shannon_entropy")

"""

- `ρ`: quantum state.

Return [Von Neumann entropy](https://en.wikipedia.org/wiki/Von_Neumann_entropy) of quantum state `ρ`.
"""
function vonneumann_entropy(ρ::Hermitian{<:Number})
    λ = eigvals(ρ)
    shannon_entropy(λ[λ .> 0])
end

vonneumann_entropy(H::AbstractMatrix{<:Number}) = ishermitian(H) ? vonneumann_entropy(Hermitian(H)) : error("Non-hermitian matrix passed to entropy")
vonneumann_entropy(ϕ::AbstractVector{T}) where T<:Number = zero(T)

"""

- `ρ`: quantum state.
- `α`: order of Renyi entropy.

Return [Renyi entropy](https://en.wikipedia.org/wiki/R%C3%A9nyi_entropy) of quantum state `ρ`.
"""
function renyi_entropy(ρ::Hermitian{<:Number}, α::Real)

    α >= 0 && α!=1 ? () : throw(ArgumentError("Parameter α must be α ≥ 0 and α ≠ 1"))
    λ = eigvals(ρ)
    λ = λ[λ .> 0]
    1/(1-α)*log(sum(λ.^α))
end

renyi_entropy(ρ::AbstractMatrix{<:Number}, α::Real) = renyi_entropy(Hermitian(ρ), α)
"""

- `ρ`: quantum state.
- `σ`: quantum state.

Return [quantum relative entropy](https://en.wikipedia.org/wiki/Quantum_relative_entropy) of quantum state `ρ` with respect to `σ`.
"""
function relative_entropy(ρ::AbstractMatrix{<:Number}, σ::AbstractMatrix{<:Number})
    log_σ = funcmh(log, σ)
    real(-vonneumann_entropy(ρ) - tr(ρ * log_σ))
end

"""

- `ρ`: quantum state.
- `σ`: quantum state.

Return [Kullback–Leibler divergence](https://en.wikipedia.org/wiki/Quantum_relative_entropy) of quantum state `ρ` with respect to `σ`.
"""
function kl_divergence(ρ::AbstractMatrix{<:Number}, σ::AbstractMatrix{<:Number})
    relative_entropy(ρ, σ)
end

"""

- `ρ`: quantum state.
- `σ`: quantum state.

Return [Jensen–Shannon divergence](https://en.wikipedia.org/wiki/Jensen%E2%80%93Shannon_divergence) of quantum state `ρ` with respect to `σ`.
"""
function js_divergence(ρ::AbstractMatrix{<:Number}, σ::AbstractMatrix{<:Number})
    0.5kl_divergence(ρ, σ) + 0.5kl_divergence(σ, ρ)
end

"""

- `ρ`: quantum state.
- `σ`: quantum state.

Return [Bures distance](https://en.wikipedia.org/wiki/Bures_metric) between quantum states `ρ` and `σ`.
"""
function bures_distance(ρ::AbstractMatrix{<:Number}, σ::AbstractMatrix{<:Number})
    sqrt(2 - 2 * fidelity_sqrt(ρ, σ))
end

"""

- `ρ`: quantum state.
- `σ`: quantum state.

Return [Bures angle](https://en.wikipedia.org/wiki/Bures_metric) between quantum states `ρ` and `σ`.
"""
function bures_angle(ρ::AbstractMatrix{<:Number}, σ::AbstractMatrix{<:Number})
    acos(fidelity_sqrt(ρ, σ))
end

"""

- `ρ`: quantum state.
- `σ`: quantum state.

Return [superfidelity](https://www.quantiki.org/wiki/superfidelity) between quantum states `ρ` and `σ`.
"""
function superfidelity(ρ::AbstractMatrix{<:Number}, σ::AbstractMatrix{<:Number})
    return real(tr(ρ'*σ) + sqrt(1 - tr(ρ'*ρ)) * sqrt(1 - tr(σ'*σ)))
end

"""

- `ρ`: quantum state.
- `dims`: dimensions of subsystems.
- `sys`: transposed subsystem.

Return [negativity](https://www.quantiki.org/wiki/negativity) of quantum state `ρ`.
"""
function negativity(ρ::AbstractMatrix{<:Number}, dims::Vector{Int}, sys::Int)
    ρ_s = ptranspose(ρ, dims, sys)
    λ = eigvals(ρ_s)
    -real(sum(λ[λ .< 0]))
end

"""

- `ρ`: quantum state.
- `dims`: dimensions of subsystems.
- `sys`: transposed subsystem.

Return [log negativity](https://www.quantiki.org/wiki/negativity) of quantum state `ρ`.
"""
function log_negativity(ρ::AbstractMatrix{<:Number}, dims::Vector{Int}, sys::Int)
    ρ_s = ptranspose(ρ, dims, sys)
    log(norm_trace(ρ_s))
end

"""

- `ρ`: quantum state.
- `dims`: dimensions of subsystems.
- `sys`: transposed subsystem.

Return minimum eigenvalue of [positive partial transposition](https://www.quantiki.org/wiki/positive-partial-transpose) of quantum state `ρ`.
"""
function ppt(ρ::AbstractMatrix{<:Number}, dims::Vector{Int}, sys::Int)
    ρ_s = ptranspose(ρ, dims, sys)
    minimum(eigvals(ρ_s))
end

"""

- `ρ`: quantum state.

Calculates the [concurrence of a two-qubit system](https://en.wikipedia.org/wiki/Concurrence_(quantum_computing)) `ρ`.
"""
function concurrence(ρ::AbstractMatrix{<:Number})
    if size(ρ, 1) != 4
        throw(ArgumentError("Concurrence only properly defined for two qubit systems"))
    end

    σ = (sy ⊗ sy)*conj.(ρ)*(sy ⊗ sy)
    λ = sqrt.(sort(real(eigvals(ρ*σ)), rev=true))
    max(0, λ[1]-λ[2]-λ[3]-λ[4])
end

1	export norm_trace, trace_distance, norm_hs, hs_distance,
2	purity, fidelity_sqrt, fidelity, gate_fidelity, shannon_entropy,
3	vonneumann_entropy, renyi_entropy, relative_entropy, kl_divergence,
4	js_divergence, bures_distance, bures_angle, superfidelity, negativity,
5	log_negativity, ppt, concurrence
6
7	"""
8
9	- `A`: matrix.
10
11	Return [trace norm](https://www.quantiki.org/wiki/trace-norm) of matrix `A`.
12	"""
13	norm_trace(A::AbstractMatrix{<:Number}) = sum(svdvals(A))	31✔
14
15	"""
16
17	- `A`: matrix.
18	- `B`: matrix.
19
20	Return [trace distance](https://www.quantiki.org/wiki/trace-distance) between matrices `A` and `B`.
21	"""
22	function trace_distance(A::AbstractMatrix{T1}, B::AbstractMatrix{T2}) where {T1<:Number, T2<:Number}	9✔
23	norm_trace(A - B) / 2	9✔
24	end
25
26	"""
27
28	- `A`: matrix.
29
30	Return [Hilbert–Schmidt norm](https://en.wikipedia.org/wiki/Hilbert%E2%80%93Schmidt_operator) of matrix `A`.
31	"""
32	norm_hs(A::AbstractMatrix{<:Number}) = sqrt(sum(abs2.(A)))	12✔
33
34	"""
35
36	- `A`: matrix.
37	- `B`: matrix.
38
39	Return [Hilbert–Schmidt distance](https://en.wikipedia.org/wiki/Hilbert%E2%80%93Schmidt_operator) between matrices `A` and `B`.
40	"""
41	function hs_distance(A::AbstractMatrix{<:Number}, B::AbstractMatrix{<:Number})	6✔
42	norm_hs(A - B)	6✔
43	end
44
45	"""
46
47	- `ρ`: matrix.
48
49	Return the purity of `ρ` ∈ [1/d, 1]
50	"""
51	purity(ρ::AbstractMatrix{<:Number}) = tr(ρ^2)	3✔
52
53	"""
54
55	- `ρ`: matrix.
56	- `σ`: matrix.
57
58	Return square root of [fidelity](https://www.quantiki.org/wiki/fidelity) between matrices `ρ` and `σ`.
59	"""
60	function fidelity_sqrt(ρ::AbstractMatrix{<:Number}, σ::AbstractMatrix{<:Number})	24✔
61	if size(ρ, 1) != size(ρ, 2) \|\| size(σ, 1) != size(σ, 2)	45✔
62	throw(ArgumentError("Non square matrix"))	3✔
63	end
64	λ = real(eigvals(ρ * σ))	24✔
65	r = real(sum(sqrt.(λ[λ.>0])))	21✔
66	end
67
68	"""
69
70	- `ρ`: matrix.
71	- `σ`: matrix.
72
73	Return [fidelity](https://www.quantiki.org/wiki/fidelity) between matrices `ρ` and `σ`.
74	"""
75	function fidelity(ρ::AbstractMatrix{<:Number}, σ::AbstractMatrix{<:Number})	6✔
76	fidelity_sqrt(ρ, σ)^2	6✔
77	end
78
79	fidelity(ϕ::AbstractVector{<:Number}, ψ::AbstractVector{<:Number}) = abs2(dot(ϕ, ψ))	6✔
80	fidelity(ϕ::AbstractVector{<:Number}, ρ::AbstractMatrix{<:Number}) = real(ϕ' * ρ * ϕ)	6✔
81	fidelity(ρ::AbstractMatrix{<:Number}, ϕ::AbstractVector{<:Number}) = fidelity(ϕ, ρ)	3✔
82
83	"""
84
85	- `U`: quantum gate.
86	- `V`: quantum gate.
87
88	Return [fidelity](https://www.quantiki.org/wiki/fidelity) between gates `U` and `V`.
89	"""
90	function gate_fidelity(U::AbstractMatrix{<:Number}, V::AbstractMatrix{<:Number})	15✔
91	abs(1.0 / size(U,1) * tr(U'*V))	15✔
92	end
93
94	"""
95
96	- `p`: vector.
97
98	Return [Shannon entorpy](https://en.wikipedia.org/wiki/Entropy_(information_theory)) of vector `p`.
99	"""
100	shannon_entropy(p::AbstractVector{<:Real}) = -sum(p .* log.(p))	24✔
101
102	"""
103
104	- `x`: real number.
105
106	Return binary [Shannon entorpy](https://en.wikipedia.org/wiki/Entropy_(information_theory)) given by \$-x \\log(x) - (1 - x) \\log(1 - x)\$.
107	"""
108	shannon_entropy(x::Real) = x > 0 ? -x * log(x) - (1 - x) * log(1 - x) : error("Negative number passed to shannon_entropy")	6✔
109
110	"""
111
112	- `ρ`: quantum state.
113
114	Return [Von Neumann entropy](https://en.wikipedia.org/wiki/Von_Neumann_entropy) of quantum state `ρ`.
115	"""
116	function vonneumann_entropy(ρ::Hermitian{<:Number})	21✔
117	λ = eigvals(ρ)	21✔
118	shannon_entropy(λ[λ .> 0])	21✔
119	end
120
121	vonneumann_entropy(H::AbstractMatrix{<:Number}) = ishermitian(H) ? vonneumann_entropy(Hermitian(H)) : error("Non-hermitian matrix passed to entropy")	21✔
122	vonneumann_entropy(ϕ::AbstractVector{T}) where T<:Number = zero(T)	3✔
123
124	"""
125
126	- `ρ`: quantum state.
127	- `α`: order of Renyi entropy.
128
129	Return [Renyi entropy](https://en.wikipedia.org/wiki/R%C3%A9nyi_entropy) of quantum state `ρ`.
130	"""
131	function renyi_entropy(ρ::Hermitian{<:Number}, α::Real)	6✔
132
133	α >= 0 && α!=1 ? () : throw(ArgumentError("Parameter α must be α ≥ 0 and α ≠ 1"))	6✔
134	λ = eigvals(ρ)	6✔
135	λ = λ[λ .> 0]	6✔
136	1/(1-α)*log(sum(λ.^α))	6✔
137	end
138
139	renyi_entropy(ρ::AbstractMatrix{<:Number}, α::Real) = renyi_entropy(Hermitian(ρ), α)	6✔
140	"""
141
142	- `ρ`: quantum state.
143	- `σ`: quantum state.
144
145	Return [quantum relative entropy](https://en.wikipedia.org/wiki/Quantum_relative_entropy) of quantum state `ρ` with respect to `σ`.
146	"""
147	function relative_entropy(ρ::AbstractMatrix{<:Number}, σ::AbstractMatrix{<:Number})	18✔
148	log_σ = funcmh(log, σ)	18✔
149	real(-vonneumann_entropy(ρ) - tr(ρ * log_σ))	18✔
150	end
151
152	"""
153
154	- `ρ`: quantum state.
155	- `σ`: quantum state.
156
157	Return [Kullback–Leibler divergence](https://en.wikipedia.org/wiki/Quantum_relative_entropy) of quantum state `ρ` with respect to `σ`.
158	"""
159	function kl_divergence(ρ::AbstractMatrix{<:Number}, σ::AbstractMatrix{<:Number})	10✔
160	relative_entropy(ρ, σ)	12✔
161	end
162
163	"""
164
165	- `ρ`: quantum state.
166	- `σ`: quantum state.
167
168	Return [Jensen–Shannon divergence](https://en.wikipedia.org/wiki/Jensen%E2%80%93Shannon_divergence) of quantum state `ρ` with respect to `σ`.
169	"""
170	function js_divergence(ρ::AbstractMatrix{<:Number}, σ::AbstractMatrix{<:Number})	3✔
171	0.5kl_divergence(ρ, σ) + 0.5kl_divergence(σ, ρ)	3✔
172	end
173
174	"""
175
176	- `ρ`: quantum state.
177	- `σ`: quantum state.
178
179	Return [Bures distance](https://en.wikipedia.org/wiki/Bures_metric) between quantum states `ρ` and `σ`.
180	"""
181	function bures_distance(ρ::AbstractMatrix{<:Number}, σ::AbstractMatrix{<:Number})	6✔
182	sqrt(2 - 2 * fidelity_sqrt(ρ, σ))	6✔
183	end
184
185	"""
186
187	- `ρ`: quantum state.
188	- `σ`: quantum state.
189
190	Return [Bures angle](https://en.wikipedia.org/wiki/Bures_metric) between quantum states `ρ` and `σ`.
191	"""
192	function bures_angle(ρ::AbstractMatrix{<:Number}, σ::AbstractMatrix{<:Number})	6✔
193	acos(fidelity_sqrt(ρ, σ))	6✔
194	end
195
196	"""
197
198	- `ρ`: quantum state.
199	- `σ`: quantum state.
200
201	Return [superfidelity](https://www.quantiki.org/wiki/superfidelity) between quantum states `ρ` and `σ`.
202	"""
203	function superfidelity(ρ::AbstractMatrix{<:Number}, σ::AbstractMatrix{<:Number})	9✔
204	return real(tr(ρ'σ) + sqrt(1 - tr(ρ'ρ)) * sqrt(1 - tr(σ'*σ)))	9✔
205	end
206
207	"""
208
209	- `ρ`: quantum state.
210	- `dims`: dimensions of subsystems.
211	- `sys`: transposed subsystem.
212
213	Return [negativity](https://www.quantiki.org/wiki/negativity) of quantum state `ρ`.
214	"""
215	function negativity(ρ::AbstractMatrix{<:Number}, dims::Vector{Int}, sys::Int)	9✔
216	ρ_s = ptranspose(ρ, dims, sys)	9✔
217	λ = eigvals(ρ_s)	9✔
218	-real(sum(λ[λ .< 0]))	9✔
219	end
220
221	"""
222
223	- `ρ`: quantum state.
224	- `dims`: dimensions of subsystems.
225	- `sys`: transposed subsystem.
226
227	Return [log negativity](https://www.quantiki.org/wiki/negativity) of quantum state `ρ`.
228	"""
229	function log_negativity(ρ::AbstractMatrix{<:Number}, dims::Vector{Int}, sys::Int)	9✔
230	ρ_s = ptranspose(ρ, dims, sys)	9✔
231	log(norm_trace(ρ_s))	9✔
232	end
233
234	"""
235
236	- `ρ`: quantum state.
237	- `dims`: dimensions of subsystems.
238	- `sys`: transposed subsystem.
239
240	Return minimum eigenvalue of [positive partial transposition](https://www.quantiki.org/wiki/positive-partial-transpose) of quantum state `ρ`.
241	"""
242	function ppt(ρ::AbstractMatrix{<:Number}, dims::Vector{Int}, sys::Int)	9✔
243	ρ_s = ptranspose(ρ, dims, sys)	9✔
244	minimum(eigvals(ρ_s))	9✔
245	end
246
247	"""
248
249	- `ρ`: quantum state.
250
251	Calculates the [concurrence of a two-qubit system](https://en.wikipedia.org/wiki/Concurrence_(quantum_computing)) `ρ`.
252	"""
253	function concurrence(ρ::AbstractMatrix{<:Number})	6✔
254	if size(ρ, 1) != 4	6✔
UNCOV 255	throw(ArgumentError("Concurrence only properly defined for two qubit systems"))	×
256	end
257
258	σ = (sy ⊗ sy)conj.(ρ)(sy ⊗ sy)	12✔
259	λ = sqrt.(sort(real(eigvals(ρ*σ)), rev=true))	6✔
260	max(0, λ[1]-λ[2]-λ[3]-λ[4])	6✔
261	end

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